Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $2x+y-z=3$, $x-y-z=\alpha$, $3x+3y+\beta z=3$ has infinitely many solution, then $\alpha +\beta -\alpha \beta$ is equal to .

Imagine you are standing in a three-dimensional room. Each of the linear equations in our system represents a flat plane stretching out infinitely in space.

Usually, three planes intersect at a single point, like the corner of a room where two walls meet the ceiling. However, in this special case, the system has infinitely many solutions. This means our planes are intersecting along a common line or lying on top of one another.

Cramer's Rule

To unlock this mystery, we turn to the elegant machinery of Cramer's Rule. For a non-homogeneous system of linear equations to possess infinitely many solutions, the main determinant of the coefficient matrix, denoted as $\Delta$, must be zero.

However, if $\Delta =0$, the system could also be inconsistent, meaning there are no solutions at all. To guarantee infinitely many solutions, we must also ensure that the determinants of the variable matrices—${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$—are all equal to zero.

This is our consistency condition, ensuring the planes are perfectly aligned to share a common space.

Let us construct our main determinant $\Delta$ using the coefficients of $x$, $y$, and $z$:

Expanding this determinant along the first row:

Simplifying the expression:

Thus, we have our first victory: $\beta =-1$.

With $\beta$ secured, we turn our attention to $\alpha$. We know that ${\Delta }_x$ must also be zero. We construct ${\Delta }_x$ by replacing the first column of our coefficient matrix with the constants from the right-hand side: $3$, $\alpha$, and $3$.

Substituting $\beta =-1$, we get:

Expanding this along the first row:

Simplifying the terms:

This leads us directly to $\alpha =3$.

We have arrived at the finish line with $\alpha =3$ and $\beta =-1$. The problem asks us to evaluate the expression $\alpha +\beta -\alpha \beta$.

Substituting our values:

The final result is $5$. You have navigated the geometry of planes and applied the rigor of Cramer's Rule to resolve the system into a clean, integer result.